Vector Network Analyzers (VNAs) have the potential to provide exceptional accuracy. One of the reasons for this exceptional performance is that a user periodically performs a calibration before making a measurement. Without proper user calibration, taking full advantage of VNA measurement accuracy is almost impossible. VNAs perform ratio-ed measurements. S-parameters Sij are defined as the ratio of the reflected or transmitted signal (through the device under test) at port i and the incident wave at port j, provided that the input signal at all other ports is zero.
Due to the increased complexity of radio frequency (RF) devices and the ongoing evolution towards differential topologies, the number of RF ports in such devices is no longer limited to two. With the advent of these new RF devices which have an increasing number of ports, there is a growing need for Vector Network Analyzers with an equally increased number of ports. This trend is further reinforced in view of the current need for simultaneous testing of several devices in production. The present invention is concerned with the calibration of such multi-port vector network analyzers.
A standard user calibration accounts for 10 or 12 sources of systematic errors being present in a VNA. Consider a forward measurement path, where the signal generator is applied to port 1 into the device under test. The first source of systematic errors is the tracking term, or the frequency response of the signal paths for the transmission and reflection measurement paths. Next is the source and load impedance match at the input and output of the device, respectively. Isolation is the small amount of leakage radiating from port 1 and being measured at port 2. Together with the coupler directivity this yields a total of six error terms in the forward path. Six similar terms obtained when the signal source is applied on port 2 for the reverse measurement path, give 12 error terms in total. If isolation can be neglected, 10 error terms remain.
Doing so allows determining raw S-parameters Sijm from the forward and reverse measurements. The forward measurements yield:S11m=b0/a0=f1(ekl,Sij)S21m=b3/a0=f2(ekl,Sij)
The reverse measurements yield:S22m=b′3/a′3=f3(e′kl,Sij)S12m=b′0/a′3=f4(e′kl,Sij)In the above expressions the quantities ekl and e′kl represent the error terms in forward and reverse direction, respectively. To obtain the calibrated S-parameters 10 or 12 error terms are applied to the above raw S-parameter measurements:Sij=F(ekl,e′kl,Sijm)
Traditionally, Short-Open-Load-Through (SOLT) is one of the best-known calibration methods at RF frequencies. It uses a well-defined short, open, and load as termination. These are referred to as calibration elements or calibration standards. One by one, each standard is connected at the reference plane and is measured by the VNA. When these steps are completed, the two reference planes are connected together to form a through connection in order to relate the error coefficients at the different ports. SOLT is sometimes also referred to as TOSM (Through-Open-Short-Match).
When extending the above to an N-port VNA, the connection of a Short, Open and Load is typically performed at each port, which is tedious and error-prone. Next a through connection is realized, at least between the first port and every other port. Other implementations require a through between each port combination. The through can either be fully known or unknown. In the latter case it is assumed to be reciprocal.
The procedure explained above maps uncalibrated S-parameters to calibrated S-parameters. Another approach starts from a Short-Open-Load (SOL)-based one-port calibration being performed at each port in combination with an error model as specified by
                              (                                                                      a                  l                                                                                                      b                  l                                                              )                =                                            K              l                        ⁡                          (                                                                    1                                                                              α                      l                                                                                                                                  β                      l                                                                                                  γ                      l                                                                                  )                                ⁢                      (                                                                                x                                          l                      ⁢                                                                                          ⁢                      1                      ⁢                                                                                          ⁢                      m                                                                                                                                        x                                          l                      ⁢                                                                                          ⁢                      2                      ⁢                                                                                          ⁢                      m                                                                                            )                                              (        1        )            This error model uses a wave formalism concept and maps the raw quantities xl1m and xl2m, measured at port 1 to the calibrated incident and reflected waves al and bl at that port. Given three known and sufficiently different terminations
      Γ    l    =            b      l              a      l      (such as Short, Open and Load) and the corresponding measured raw quantities, one simply needs to solve a set of linear equations to retrieve α1, β1 and γ1. Sufficiently different refers to the fact that the different terminations must allow extracting the error coefficients α1, β1 and γ1 in a numerically well-conditioned way.
Next the Kl parameters can be extracted by connecting a known or an unknown reciprocal through between port 1 and port l. However, using only a relative calibration, one of the parameters Ki cannot be determined and typically it is set to unity at port 1. The latter is not an issue in order to obtain calibrated ratios such as S-parameters. For this purpose alm and blm are defined as
                              (                                                                      a                  lm                                                                                                      b                  lm                                                              )                =                                                            (                                                                            1                                                                                      α                        l                                                                                                                                                β                        l                                                                                                            γ                        l                                                                                            )                            ⁢                              (                                                                                                    x                                                  l                          ⁢                                                                                                          ⁢                          1                          ⁢                                                                                                          ⁢                          m                                                                                                                                                                        x                                                  l                          ⁢                                                                                                          ⁢                          2                          ⁢                                                                                                          ⁢                          m                                                                                                                    )                                      ⇒                          (                                                                                          a                      l                                                                                                                                  b                      l                                                                                  )                                =                                    K              l                        ⁡                          (                                                                                          a                      lm                                                                                                                                  b                      lm                                                                                  )                                                          (        2        )            By performing a measurement in the forward (indicated by superscript F) and reverse direction (indicated by superscript R) while inserting a known through or an unknown reciprocal through between port 1 and port 2, K2 can be extracted using:
                              (                                                                      b                  1                                                                                                      b                  2                                                              )                =                                                            (                                                                                                    S                        11                                                                                                            S                        12                                                                                                                                                S                        12                                                                                                            S                        22                                                                                            )                            ⁢                              (                                                                                                    a                        1                                                                                                                                                a                        2                                                                                            )                                      ⇒                          (                                                                                          b                                              1                        ⁢                                                                                                  ⁢                        m                                                                                                                                                        b                                              2                        ⁢                                                                                                  ⁢                        m                                                                                                        )                                =                                    (                                                                                          S                      11                                                                                                                                                    K                          2                                                                          K                          1                                                                    ⁢                                              S                        12                                                                                                                                                                                                          K                          1                                                                          K                          2                                                                    ⁢                                              S                        12                                                                                                                        S                      22                                                                                  )                        ⁢                          (                                                                                          a                                              1                        ⁢                                                                                                  ⁢                        m                                                                                                                                                        a                                              2                        ⁢                                                                                                  ⁢                        m                                                                                                        )                                                          (        3        )            from which is obtained
                              (                                                                      b                                      1                    ⁢                                                                                  ⁢                    m                                    F                                                                              b                                      1                    ⁢                                                                                  ⁢                    m                                    R                                                                                                      b                                      2                    ⁢                                                                                  ⁢                    m                                    F                                                                              b                                      2                    ⁢                                                                                  ⁢                    m                                    R                                                              )                =                              (                                                                                S                    11                                                                                                                                      K                        2                                                                    K                        1                                                              ⁢                                          S                      12                                                                                                                                                                                      K                        1                                                                    K                        2                                                              ⁢                                          S                      12                                                                                                            S                    22                                                                        )                    ⁢                      (                                                                                a                                          1                      ⁢                                                                                          ⁢                      m                                        F                                                                                        a                                          1                      ⁢                                                                                          ⁢                      m                                        R                                                                                                                    a                                          2                      ⁢                                                                                          ⁢                      m                                        F                                                                                        a                                          2                      ⁢                                                                                          ⁢                      m                                        R                                                                        )                                              (        4        )            
In the case of an unknown through, solving the above equation provides S11, S22,
      S    12    ′    =                    K        2                    K        1              ⁢          S      12      and
      S    21    ′    =                    K        1                    K        2              ⁢                  S        12            .      Hence S12=√{square root over (S′12·S′21)}, which requires proper root selection. The latter can be found by modelling the through as a lossy delay, by phase unwrapping or any other suitable approach. Finally K2 is obtained as S′12/S12 keeping in mind that K1 was chosen to be unity. This procedure can be repeated for all remaining ports and is as such easily scalable with respect to the number of ports.
However, the above approach requires a considerable number of human interventions to connect the different calibration elements to the respective ports. Furthermore this process is error-prone because of the possibility to connect another calibration element than the requested or assumed one.